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Understanding Randomness

What is it about chance outcomes being random that makes random selection seem fair? Two things: Nobody can guess the outcome before it

happens. When we want things to be fair, usually some

underlying set of outcomes will be equally likely (although in many games some combinations of outcomes are more likely than others).

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Example: Pick “heads” or “tails.” Flip a fair coin. Does the outcome match your

choice? Did you know before flipping the coin whether or not it would match?

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Statisticians don’t think of randomness as the annoying tendency of things to be unpredictable or haphazard.

Statisticians use randomness as a tool. But, truly random values are surprisingly

hard to get…

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It’s surprisingly difficult to generate random values even when they’re equally likely.

Computers have become a popular way to generate random numbers. Even though they often do much better than

humans, computers can’t generate truly random numbers either.

Since computers follow programs, the “random” numbers we get from computers are really pseudorandom.

Fortunately, pseudorandom values are good enough for most purposes.

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There are ways to generate random numbers so that they are both equally likely and truly random.

The best ways we know to generate data that give a fair and accurate picture of the world rely on randomness, and the ways in which we draw conclusions from those data depend on the randomness, too.

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We need an imitation of a real process so we can manipulate and control it.

In short, we are going to simulate reality.

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The sequence of events we want to investigate is called a trial.

The basic building block of a simulation is called a component. Trials usually involve several components.

After the trial, we record what happened—our response variable.

There are seven steps to a simulation…

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1. Identify the component to be repeated.2. Explain how you will model the component’s

outcome.3. Explain how you will combine the components to

model a trial.4. State clearly what the response variable is. 5. Run several trials.6. Collect and summarize the results of all the trials.7. State your conclusion.

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Don’t overstate your case. Beware of confusing what really happens with

what a simulation suggests might happen. Model outcome chances accurately.

A common mistake in constructing a simulation is to adopt a strategy that may appear to produce the right kind of results.

Run enough trials. Simulation is cheap and fairly easy to do.

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How to harness the power of randomness. A simulation model can help us investigate a

question when we can’t (or don’t want to) collect data, and a mathematical answer is hard to calculate.

How to base our simulation on random values generated by a computer, generated by a randomizing device, or found on the Internet.

Simulations can provide us with useful insights about the real world.

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The game of 21 can be played with an ordinary 6-sided die. Competitors each roll the die repeatedly, trying to get the highest total less than or equal to 21. If your total exceeds 21, you lose.

Suppose your opponent has rolled an 18. Your task is to try to beat him by getting more than 18 points without going over 21. How many rolls do you expect to make, and what are your chances of winning?

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